14. Low Rank Changes in A and Its Inverse

TL;DR

The video discusses low-rank matrices, explores a rank 1 perturbation of the identity matrix, introduces the Sherman-Morrison-Woodbury formula, and highlights its applications in solving linear systems and in the Kalman filter.

Transcript

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GILBERT STRANG: So just to orient where we are, today s... Read More

Key Insights

• 😘 Low-rank matrices can be truly low-rank or approximately low-rank, depending on the nature of their singular values.
• 👻 The Sherman-Morrison-Woodbury formula allows for the inversion of an n by n matrix using a simpler 1 by 1 matrix inverse.
• 😜 Perturbing a matrix by rank 1 results in a corresponding rank 1 change in its inverse.

Questions & Answers

Q: What is the difference between truly low-rank matrices and approximately low-rank matrices?

Truly low-rank matrices have a rank of 1, while approximately low-rank matrices have singular values that decrease rapidly.

Q: How does the Sherman-Morrison-Woodbury formula simplify the process of inverting matrices?

The formula allows for the inversion of an n by n matrix in terms of the inverse of a simpler 1 by 1 matrix, resulting in a more manageable computation.

Q: Why is the rank 1 perturbation of the identity matrix significant?

The rank 1 perturbation of the identity matrix demonstrates that changing a matrix by rank 1 corresponds to a rank 1 change in its inverse.

Q: What are some practical applications of the Sherman-Morrison-Woodbury formula?

The formula is commonly used in solving linear systems and implementing the Kalman filter for dynamic least squares problems.

Summary & Key Takeaways

• The video begins by introducing low-rank matrices and differentiating between truly low-rank matrices and approximately low-rank matrices.

• The rank 1 perturbation of the identity matrix is considered, and the video explores the inverse of this perturbed matrix using the Sherman-Morrison-Woodbury formula.

• Several key insights are presented, including how changing a matrix by rank 1 results in a corresponding rank 1 change in its inverse.

• The video delves into the practical applications of the Sherman-Morrison-Woodbury formula, such as solving linear systems and implementing the Kalman filter for dynamic least squares problems.